A335583 Number of 1's in the partitions of n into 4 parts.

0, 0, 0, 0, 4, 3, 5, 6, 8, 9, 12, 13, 16, 18, 21, 23, 27, 29, 33, 36, 40, 43, 48, 51, 56, 60, 65, 69, 75, 79, 85, 90, 96, 101, 108, 113, 120, 126, 133, 139, 147, 153, 161, 168, 176, 183, 192, 199, 208, 216, 225, 233, 243, 251, 261, 270, 280, 289, 300, 309, 320, 330, 341

Offset

0,5

Links

Andrew Howroyd, Table of n, a(n) for n = 0..10000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} [k = 1] + [j = 1] + [i = 1] + [n-i-j-k = 1], where [ ] is the Iverson bracket.

From Andrew Howroyd, Nov 25 2025: (Start)

G.f.: 4*x^4 + 3*x^5/(1 - x) + 2*x^6/((1 - x)*(1 - x^2)) + x^7/((1 - x)*(1 - x^2)*(1 - x^3)).

G.f.: x^4*(4 - x - 2*x^2 - 2*x^3 + x^4 + 2*x^5 - x^6)/((1 - x)^3*(1 + x)*(1 + x + x^2)). (End)

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - Wesley Ivan Hurt, Nov 28 2025

G.f.: x^4 * Sum_{j=0..3} 1 / Product_{k=1..j} (1 - x^k). - Seiichi Manyama, May 13 2026

a(n) = [n>3]*floor((n^2 + 4*n + 6*[(n mod 6) = 4])/12 + [n<5]), where [ ] is the Iverson brackets. - Hoang Xuan Thanh, May 17 2026

Mathematica

Table[Sum[Sum[Sum[KroneckerDelta[k, 1] + KroneckerDelta[j, 1] + KroneckerDelta[i, 1] + KroneckerDelta[n - i - j - k, 1], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]

Prog

(PARI) seq(n) = Vec(x^4*(4 - x - 2*x^2 - 2*x^3 + x^4 + 2*x^5 - x^6)/((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x*x^n), -n-1); \\ Andrew Howroyd, Nov 25 2025

Crossrefs

Column k=4 of A394828.

Sequence in context: A296413 A016701 A023829 * A328109 A000211 A059902

Adjacent sequences: A335580 A335581 A335582 * A335584 A335585 A335586

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 26 2021

Status

approved